It is then surprising that, despite the vast amount of theoretical [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]} and experimental [12]}, [13]}, [14]}, [15]}, [16]}, [17]} researches on the helical edge states, an explicit closed-form analytic expressions for different properties of such kind of edge states (e.g. wave functions, dispersion relations, and spin rotations) are still lacking.
Analytical investigations so far have been restricted to the use of a combination of analytical and numerical methods. For instance,
the chiral edge states of the Haldane model are studied in Ref. [2]} by using the Harper equation to find the wave function transfer relation between two edges in a graphene ribbon. The same method is used to study the edge states of the Kane-Mele model for similar systems [3]}.
